After Dashrath's death, Bharat come with his army to lord Rama and to take lord Ram back to Ayodhya.
After doing necessary rituals for their father's soul, there was a meeting where Bharat brought up some arguments which he wants to say to lord Rama. In that meeting,

What arguments did Bharat and Javali/Jawali (a Brahman sitting in the meeting breaking the silence of the meeting states some arguments) state before lord Ram on occasion to "Shri Ram to return and rule the kingdom"?

And what are the words replied by lord Rama to each of them. Did lord Rama's words were based on Vedas?

1 Answer
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In this chapter of the Ayodhya Kanda of the Ramayana, Bharata uses many arguments to try to persuade Rama to take the throne. He says that Rama's place is in Ayodhya ruling the kingdom, not the forest, and that Rama should reverse the mistakes of Dasharatha and Kaikeyi.

Who, in this world, is endowed with such qualities like you, O annihilator of enemies! Neither does sorrow disturb you nor does joy enrapture you. You are highly honoured by elders and you clarify your doubts from them. By what should he be moved, who looks on life and death, misfortune or prosperity with an equal eye? O, King! He who knows the totality of time as you do, ought not to feel despondent even on meeting with adversity. O, the magnanimous Rama! you are equally strong as celestials. You are true to your promise, all-knowing, all-viewing and endowed with understanding. Even the most fearful calamity ought not to attack you, endowed as you are, with such attributes; and recognizing the birth and end of all beings. The sinful deed, committed by my mean mother for my sake during my absence, was not to my liking. (Hence) be gracious to me. I am bound by the chains of morality. For that reason, I do not kill with a severe punishment, my mother who was of perverse conduct and worthy of retribution. How can I, born of Dasaratha of noble birth and deeds and conversant of right and wrong, do the most wicked and horrible act? The King Dasaratha was my preceptor, had several sacrificial performances to his credit was aged, was my father, equal to a deity and is now dead. Hence, I cannot reproach him in an open assembly. O Rama the knower of piety! What man who knows the principles of righteousness, would commit such an improper act, bereft of one's interest and duty, in order just to please a woman? There is an ancient saying that at the time of death, beings lose their judgment. The king, acting as he did, has made this saying indeed true in the eyes of the world. This transgression of righteousness was done by our father for fear of anger (of Kaikeyi) or through infatuation and over-haste. You do keep in view a noble purpose and reverse that transgression. A son who honours only a good deviation (from righteousness) of a father is accepted as a real son in this world. He who acts otherwise than this would be facing a quite reverse situation. You be that real son. Do not lend yourself to the infamous act, committed by our father, which in this world, confident men freely condemn. Save Kaikeyi and me, our father, our friends, our relatives, all the urbanites and the villagers as well as the entire kingdom. Where is the forest and where is the princely dignity? Where is the matted hair and where is the protection of a kingdom? You ought not to do such a contradictory act. O, the highly enlightened hero! The first duty of a warrior is indeed to get crowned, so that he may then protect his subjects. Which man of warrior-caste would set aside his manifest duty and practise a virtue that gives doubtful result, which does not promise happiness, which has a mental restraint and which is uncertain? If you desire to pursue a virtue alone; although born of pain, then take pains while protecting the four castes as per tradition, by recourse to righteousness. O, Rama the knower of righteousness! The knower of what is right indeed tell that the life of a house-holder is the noblest of all the four stages of life. How can you abandon it? Four Stages of life Brahmacharya (student-life); Garhasthya (life of a house-holder); Vanaprastha (life of a hermit); Sannyaasa (Life of an ascetic). I am indeed junior to you in the point of learning, situation and by birth. How should I rule the earth, while still you are living? I, a mere child, who am devoted of intelligence and virtues besides placed in an inferior position, cannot even live without you. O, Rama the knower of righteousness! Rule this entire kingdom belonging to your father, without distraction, without hindrance and as per the conduct prescribed for you along with your relatives. Let all the king's ministers along with Vasishta and other priests well-versed in sacred formulas, crown you, the knower of sacred texts, here itself. You, having been crowned thus, may proceed to Ayodhya along with us to rule it, like Indra the Lord of celestials, having conquered all the worlds with his might, proceeds to his kingdom along with celestials. You shall discharge the three obligations subdue all the enemies, gratify the wishes of all your friends and you alone rule me in Ayodhya. Three obligations to the Gods, the Ancestors and Brahmins. O, excellent among men! Wipe off my mother's curse today and protect our venerable father from his sin. I request you by bowing my head before you. Show compassion on me and on all our relatives as Lord Shiva on all beings. Else if you disregard my entire request and move again into the forest form here, I too will go along with you.

In the next chapter, Rama responds by saying he is duty-bound to fulfill Dasharatha's promise:

These words, which you have spoken are worthy of you, the son of Dasaratha the excellent king, born through Kaikeyi. O, My brother! Long ago, when our father married your mother, he promised your maternal grandfather that he would confer his kingdom as an exceptional marriage-dowry. Thereafter, in a conflict between Gods and demons, your mother received the promise of two boons from the efficient lord of the earth, King Dasaratha, as a token of his joy and gratitude. O, Tiger among men! Your illustrious mother of beautiful complexion consequently demanded these two boons from that chief of men, for you the throne and for me the exile to the forest. O, excellent among men! I too, have been enjoined by our aforesaid father to live here in the forest for fourteen years, in accord with granting of boon. I as such, without any rival, have come to this lonely forest accompanied by Lakshmana and Seetha in order to carry out the promise given by our father. You too ought, likewise, to make our father, as a person having given a true promise, O Indra (the Lord of Celestials) among kings, by getting yourself anointed to the crown without any delay. O, Bharata! For my sake relieve the mighty king from his vow and make both our mother and father happy. My dear brother! Formerly, an illustrious king named Gaya, while performing a sacrifice in a place called Gaya in honour of his ancestors, chanted the following verse: "Since a son delivers his father from a place of torment (hell) called 'Put', he is named as 'Putra'- 'he who delivers his ancestors from all dangers' To have many virtuous and learned sons is to be desired, since one, atleast among them, who is intimately connected will come to Gaya to perform a sacrifice. O, prince! This is the conviction of all the royal sages. O, the efficient and the excellent of men! Therefore, save our father from hell. O, the valiant Bharata! Go to Ayodhya along with Shatrughna and all the Brahmanas and give joy to the people there. O, King! I too, without delay, will proceed to Dandaka forest along with Seetha and Lakshmana. O, Bharata! You become the Lord of men. I will become the emperor of the wild beasts of the forest! Return now to the excellent city of Ayodhya full of joy and I also full of joy will enter Dankada Forest! O, Bharata! Let the (royal) white umbrella provide a cool shadow for your head, repulsing the rays of sunlight. I will take shelter comfortably under the abundant shadow of these forest-trees. O, Bharata! The wise Shatrughna is helpful to you. Lakshmana is known to be an outstanding friend for me. We, the four excellent sons, will ultimately make the king true to his promise. Do not get disappointed.

In the next chapter, Dasharatha's minister Jabali presents Charvaka-like arguments that Rama should forget aboutDharma and just pursue a materialistic lifestyle in Ayodhya:

Enough, O Rama! Let not your wisdom be rendered void like a common man, you who are distinguished for your intelligence and virtue. Who is related to whom? What is there to be obtained by anything and by whom? Every creature is born alone and dies alone. O, Rama! He who clings to another, saying, 'This is my father, this is my mother, he should be known as one who has lost his wits. There is none who belongs to another. O, Rama! As one who passes the a strange village spends the night the and the next day leaves that place and continues his journey, so are mother, father, home and possessions to a man; they are but a resting place. The wise do not become attached to them. O, chief of men! You as such should not abandon your father's kingdom in order to dwell in a lonely forest, that is excruciating hard to traverse and full of thorny thickets. Get yourself crowned in the prosperous kingdom of Ayodhya. That city is waiting for you, with your locks duly unfound. O, prince! Enjoy the royal luxuries worthy of you. Move around in Ayodhya as Indra the Lord of celestials does in heaven! Dasaratha is none to you nor you in anyway to him. That king is another and your are another. Hence, do what is told by me? The father is only the seed of a being. The sperm and the ovum blend at the right time in the mother's womb, so that a human being is born in this world. The king has gone, where he had to go. This is the fate of all being unnecessarily, you are still frustrated over the matter. I pity all those whosoever, devoted to wealth and religious merit, not other (who are devoted to sense-enjoyment), for, they, having undergone suffering in this life, have met with extermination after death. These people say, 'The eighth day should be given up to sacrifices for the spirits of our ancestors.' See the waste of food. What will a dead man eat? If food eaten by one here, reaches another's body, then let a sacrifice be offered for those who are setting out on a distant journey. Will it not become a food on their path? 'Perform sacrifices, distribute gifts, consecrate yourselves, practise austerity and renunciation' - These writings are composed by learned men for the sake of inducing others to give. O, the highly wise! Arrive at a conclusion, therefore, that there is nothing beyond this Universe. Give precedence to that which meets the eye and turn your back on what is beyond our knowledge. Honour the judgment of the wise and regarding that which is approved by all, accept the kingdom as propitiated by Bharata.

In the next chapter, Rama gives a defense of Dharma and a rebuke of Jabari, saying that he regrets that Dasharatha ever hired a Charvaka like Jabali:

That which, in your desire to please me, you have counselled, is impossible though it appears to be possible. It is like a forbidden food, appearing to be an agreeable one. He who is unruly, fully invested with ill-conduct, having a bad reputation and seeing differences in everything, does not gain respect from honest men. One's conduct itself explains whether one belongs to a good family or a bad family, valiant or arrogant and chaste or unchaste. What sensible man, able to discern what is just and what is unjust, in this world, would respect me, if I am ignoble resembling as noble, bereft of honesty, impure, having no good qualities but appearing like the one having good qualities, ill-behaved but appearing as well-behaved abandoning righteousness and getting hold of unrighteousness in the guise of piety, creating confusion in the world and disregarding rules of conduct. If I behave in this manner faithlessly, to whom can I advise a prescribed conduct? How would I attain heaven? This entire world would follow its own whims, for, whatever the conduct of the kings may be, such will be the conduct of their subjects. The eternal royal governance is indeed an assemble of truth and not cruel. Hence, the kingship has the Truth as its essence. The world is established in Truth. Even sages and divine being have respected truthfulness alone. The one who speaks truth obtains the highest position in this world. People fear of a person, who speaks untruth, as one fears a snake. Truth is the highest virtue and is stated to be the origin of heaven. Truth is god and all virtues follow truth. All are rooted in truth there is nothing higher than truth. Gift sacrifice, oblation, austerities performed and the scriptural texts have the foundation in Truth. Hence, one should thoroughly surrender to truth. One rules over the world. One develops a race. One sinks into hell . One rises high to heaven (according to one's degree of truthfulness practiced). I am true to my promise. Why should I not fulfill the command of my father, who was a devotee of truth? Neither covetousness nor forgetfulness nor pride would cause me to destroy the bond of morality. I shall honour the vow made to my father. Neither gods nor the manes will accept the offerings of those who are wanting in truth, unsteady and unstable in their minds. This is what is taught to us. I perceive this virtue i the form of truthfulness as a universal permeation of spirit. That is why, this burden, observed as a vow, has been honoured by good men. I renounce the so-called duty of a warrior, it is injustice under the name of justice, it is practised by petty cruel and covetous men of evil deeds. Sin is committed by the body after it has been conceived by the mind and falsehood is spoken with the tongue. Thus, the degrading act is of three types (with body, mind and tongue) The earth, fame, prosperity and fortune indeed woo a man of truth. They constantly oblige the truth: truth should therefore be strictly observed! The logical words, you have made it out to be good, saying 'Do this good thing' as uttered to me, are really unworthy. Having promised before my father about my exile to the forest, how can I fulfill Bharata's words now, abandoning the father's words? A firm promise has been made by me in the presence of my father, when Queen Kaikeyi too became rejoiced. I will proceed with life-journey in this manner, by accepting this dwelling in the forest, by remaining pure in body and mind, having controlled my diet, by feasting the Gods and Ancestors with pure roots, flowers and fruits, with all my five senses fully sated, without any deceit, fully devout and discriminative of what ought to be done and what ought not to be done. Having reached this terrestrial globe (the realm of action), only a virtuous act is to be undertaken. The god of fire, the wind-god and the moon-god reap the fruits of their acts. Having performed a hundred sacrifices, Indra the Lord of celestials went to heaven. Having practised severe austerities, the great sages went to heaven. The virtuous say that truth, piety, valour, compassion for all beings, polite speech and worship of Brahmanas, gods and unexpected guests are the paths to heaven. Therefore the learned, well-instructed in what is to be their greatest advantage; follow their purpose resolutely and fulfill their duty in its entirety properly and attentively, seeking to attain the highest realms. I accuse the act done by my father in taking you into his service, you with your misleading intelligence, a firm atheist fallen from the true path. It is an exact state of the case that a mere intellection deserves to be punished as it were a thief and know an atheist to be on par with a mere intellectual. Therefore he is the most suspectable and should be punished in the interest of the people. In no case should a wise man consort with an atheist. Men that lived before you have performed many auspicious acts, abandoning all hopes of reward in this world as well as the next. Therefore, Brahmanas offer oblations in the sacred fire and practise noble deeds. The sages who are devoted to righteousness, associating with men of virtue, endowed with spiritual splendour, practising abundant charity, harm-less and washed free of all taint, are honoured into the world.

But Jabali explains that he was just pretending to profess Charvaka beliefs just to try to persuade Rama to return to Ayodhya.